Why do this problem?
This problem
provides an interesting situation in which to consider and
practise the formula for the sum of an arithmetic progression
with common difference $1$ (although the knowledge of APs is not
necessary). Relating the sums to a visual representation on a
number line will reinforce the meaning of the algebraic process
involved in the derivation of the AP formula.
Possible approach
Ask students to start the problem by finding the APs leading to
$544$ and $424$. Once they have represented their sums on the
number line, can they explain in words why the sums will yield
$544$ and $424$ without calculation? They can use the AP
formula to check their answers (they will need calculators).
Next students can look at the numbers $1000$ and $1001$. Again,
can they explain in words using a representation on a number
line why their answers work? They can use the AP formula to
check that they are indeed correct.
In searching for the APs, students should realise that the
factorisation of a number is important in breaking it down into
an AP. They should realise or be encouraged to relate the
factors of a number to the terms $a$ and $n$ of the
arithmetical progression. You might suggest that students also
try to find APs for other numbers such as $40, 246$ and $500$.
Finally, note that there are two possible ways of writing the
formula for the sum of an AP
$$S_n = \frac{n}{2}\left[2a+(n-1)\right] =
n\left[a+\frac{(n-1)}{2}\right]$$
In words, what are the equations saying? Which of these two
representations would be useful for different polite numbers?
In order to prove the case that a particular number is polite,
students will need to show that it can be written in one of the
above algebraic forms: essentially, they need to try to break
down each number into two appropriate factors.
Proving the final part will first require a conjecture on the
part of the students. Before attempting to prove their result
they might be advised to test out their conjecture on some small
numbers less than $40$. This conjecturing is an important part of
the mathematical process, irrespective of whether or not a final
proof is constructed.
Key questions
Can you explain in words why your consecutive sum will yield
the required answer? Could you see how this approach might work
for other numbers?
Why is the factorisation of the numbers important?
What is your conjecture for which numbers are polite?
Possible extension
Which numbers can be written as the sum of an arithmetical
progression with common difference $2$?
Possible support
Some students may find it hard to find the APs leading to $544$
and $424$ straight away, and may benefit from experimenting
with smaller numbers, perhaps using the approaches suggested in
Consecutive Sums . This should help them to spot some
patterns and make some conjectures, which can then be
investigated using the AP formula.